CHI-SQUARE TEST EXPLANATION

Let’s say you want to know if there is a difference in the proportion of men and women who are left handed and let’s say in your sample 10% of men and 5% of women were left-handed.

How it’s Calculated (Without the gory details)

1. You collect the data. For example, you ask 120 men and 140 women which hand they use and get this:

|ACTUAL DATA |Left-handed |Right-handed |
|Men |12 |108 |
|Women |7 |133 |

2. Calculate what numbers of left and right-handers we would expect IF men and women were the same. In this case, IF men and women were equally left and right handed, we would have expected these numbers in our sample of 260 people (Ask if you want to know how this is done):

|EXPECTED IF NO DIFFERENCE |Left-handed |Right-handed |
|Men |8.77 |111.23 |
|Women |10.23 |129.77 |

3. The computer calculates a Chi-square (pronounced Ki-square) value. The Chi-square value is a single number that adds up all the differences between our actual data and the data expected if there is no difference. If the actual data and expected data (if no difference) are identical, the Chi-square value is 0. A bigger difference will give a bigger Chi-square value.

4. Look up the Chi-square value in a table to see if it is big enough to indicate a significant difference in handedness of males and females.

Interpretation
Greater differences between expected and actual data produce a larger Chi-square value. The larger the Chi-square value, the greater the probability that there really is a significant difference.

With a 2 by 2 table like this (If you have more than 4 cells of data in your table, see your instructor):

If the Chi-square value is greater than or equal to the critical value
There is a significant difference between the groups we are studying. That is, the difference between actual data and the expected data (that assumes the groups aren’t different) is probably too great to be attributed to chance. So we conclude that our sample supports the hypothesis of a difference.

If the Chi-square value is less than the critical value
There is no significant difference. The amount of difference between expected and actual data is likely just due to chance. Thus, we conclude that our sample does not support the hypothesis of a difference.

In this example, the critical value is 3.8. The Chi-square value was 2.383, which is less than 3.8. Thus, there is no significant difference in handedness between men and women in our sample. We conclude that based on this sample, men and women in general seem equally likely to be left or right handed.

**Warning**
We have not proven anything!!! These first samples might be atypical. Repeated sampling may show a significant difference, or eliminate the difference we thought we saw. Because of this uncertainty, we can only say that the hypothesis was supported or not supported.

Graphing Categorical Data that you analyze with the Chi-Square Statistic

Let’s say you want to know if there is a difference in the proportion of men and women who are left handed and you collect this data:

|ACTUAL DATA |Left-handed |Right-handed |
|Men |12 |108 |
|Women |7 |133 |

Calculate the % of men that are left-handed and right handed in your study: 12 / 120 total men = 0.10 X 10 = 10% of men were left-handed 108 / 120 = 0.90 X 100 = 90% of men were right-handed

Do the same for women: 7 / 140 total women = 0.05 X 100 = 5% left handed 133 / 140 = 0.95 X 100 = 95% right-handed

Plot these percents as bars:

Or plot as stacked bars:

Or plot as a pie diagram.